Vol. 9, No. 2, 2016

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ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
The isoperimetric and Kazhdan constants associated to a Paley graph

Kevin Cramer, Mike Krebs, Nicole Shabazi, Anthony Shaheen and Edward Voskanian

Vol. 9 (2016), No. 2, 293–306
Abstract

In this paper, we investigate the isoperimetric constant (or expansion constant) of a Paley graph, and the Kazhdan constant of the group and generating set associated with a Paley graph.

We give two new upper bounds for the isoperimetric constant h(Xp) for the Paley graph Xp. These bounds improve previously known eigenvalue bounds on h(Xp). Along with a known eigenvalue lower bound for h(Xp), they provide a narrow strip in which h(Xp) must live. More precisely, we show that (p p)4 h(Xp) (p 1)4, which implies that limph(Xp)p = 14.

In addition, we show that the Kazhdan constant associated with the integers modulo p and the generating set for the Paley graph Xp approaches 2 as p tends to infinity, which is the best possible limit that the Kazhdan constant can be.

Keywords
isoperimetric constant, expansion constant, Paley graph, Kazhdan constant
Mathematical Subject Classification 2010
Primary: 05C99
Milestones
Received: 28 October 2014
Revised: 25 January 2015
Accepted: 2 February 2015
Published: 2 March 2016

Communicated by Kenneth S. Berenhaut
Authors
Kevin Cramer
Department of Mathematics
California State University, Los Angeles
Los Angeles, CA 90032
United States
Mike Krebs
Department of Mathematics
California State University, Los Angeles
Los Angeles, CA 90032
United States
Nicole Shabazi
Department of Mathematics
California State University, Los Angeles
Los Angeles, CA 90032
United States
Anthony Shaheen
Department of Mathematics
California State University, Los Angeles
Los Angeles, 90032
United States
Edward Voskanian
Department of Mathematics
University of California, Riverside
Riverside, CA 92521
United States